Solve x^2+19x-20

$${ x }^{ 2 } +19x-20$$

Factor

$$\left(x-1\right)\left(x+20\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-20. To find a and b, set up a system to be solved.

$$a+b=19 ab=1\left(-20\right)=-20$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -20.

$$-1,20 -2,10 -4,5$$

Calculate the sum for each pair.

$$-1+20=19 -2+10=8 -4+5=1$$

The solution is the pair that gives sum 19.

$$a=-1 b=20$$

Rewrite x^{2}+19x-20 as \left(x^{2}-x\right)+\left(20x-20\right).

$$\left(x^{2}-x\right)+\left(20x-20\right)$$

Factor out x in the first and 20 in the second group.

$$x\left(x-1\right)+20\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+20\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}+19x-20=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-20\right)}}{2}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-20\right)}}{2}$$

Multiply -4 times -20.

$$x=\frac{-19±\sqrt{361+80}}{2}$$

Add 361 to 80.

$$x=\frac{-19±\sqrt{441}}{2}$$

Take the square root of 441.

$$x=\frac{-19±21}{2}$$

Now solve the equation x=\frac{-19±21}{2} when ± is plus. Add -19 to 21.

$$x=\frac{2}{2}$$

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-19±21}{2} when ± is minus. Subtract 21 from -19.

$$x=-\frac{40}{2}$$

Divide -40 by 2.

$$x=-20$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -20 for x_{2}.

$$x^{2}+19x-20=\left(x-1\right)\left(x-\left(-20\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}+19x-20=\left(x-1\right)\left(x+20\right)$$

Evaluate

$$\left(x-1\right)\left(x+20\right)$$